Beam Dimension and concrete formula 2

The spacing of the piers depends on the bearing capacity of each pier at your specific location as well as the maximum span that your suspended floor can provide. The concrete beam dimensions can be a multitude of sizes depending on how much rebar you want to buy...i.e. less concrete depth = more rebar.

18" diameter piers x 25' deep would appear to have a huge amount of capacity but this depends on the soil. Do you have a geotechnical engineer making these recommendations?

There really isn't a specific answer to your question without really getting into the project. If you have trouble with this, you might want to hire a structural engineer to actually give you an economical design - well worth the SE's fee I'm sure.

Recomendation for pier dimensions came as a result of GeoTech report. Highly expansive clay soils are present to depths greater than 30'.
The location is Pottsboro Texas. In my mind an 18"W X 12"H beam would create the necessary crawl space and support structure for brick ledge and 2X10 floor joists which span the 15' distance to beam running down midline of house.
In my hand drawn diagram I have the piers spaced approximately 7'.
I have noted others in area using 2 5/8" rebar on bottom and top of beams but these examples were soil supported slab on grade. I don't have access to any graph's or formulae and wouldn't know how to calculate needed dimensions and needed PSI of concrete and rebar to support the structure and span piers.
I'll hire an SE to design foundation but I need to estimate cost of this custom home for lending institution.

The spacing of the piers is dependent on many things, but primarily on vertical and lateral load capacity. Is this a sloping site? Is there a known tendency for homes in your area to experience differential settlement (do your neighbors complain of their houses leaning strangely, and perhaps differnetly in the summer than the winter?), is there a body of water nearby?

The grade beams must be designed by an engineer and should be considered as though there is no soil below it. This is because of the fact that the expansive soil may contract and provide no bearing capacity. If you are on a sloping site, you may have lateral loads imposed on the piers as well. I don't know much about Texas seismic requirements, but te design of the foundation may use the piers for lateral resistance of earthquake forces or that may be left to the grade beams if the loads are small. In this case, the beams would have to be designed for this additional load.

In short there are numerous ways that a foundation of this sort can be designed...you really need an engineer to review this.

I have discussed the design of the foundation with an SE. He has been to the site and reviewed the Geo-Tech report and his recommedation is to reduce the pier size and depth. He will design the complete foundation system and explain his design strength requirements for the lenders.

The SE has designed the foundation. The piers are 12" dia. X 15' deep with a 24" dia. bell on the bottom. Concrete is to be 3500psi. A steel reinforcing cage is installed in each pier. 15' depth is below active zone of expansive clay soil. Piers are spaced at 6 feet.
Perimeter grade beam is 14"W X 24"H and has 4 #5 rebar running through it. Rebar of beam are tied into reinforcing cage of each pier and the concrete (2nd pour)of the perimeter grade beam encapsulates both reinforcements.
14" perimeter grade beam allows for brick ledge, and 8" concrete blocks. Floor joists are installed on top of concrete blocks bringing total installed height of all elements to the desired design elevation of the main floor.
The SE has two rows of interior piers running the long dimension (60 feet) of the building spaced at 6'. 4 X 4 wooden beams span those piers. 2 X 8 floor joists are placed across the 4 X 4 wooden beams joining the perimeter beams on the outside.
My question is: can I save time, material, and money by having (one) row of piers spaced at 10' spanned by a wooden beam of triple 2 x 10s (nailed together) running down middle of long dimension of building with 2 X 10 floor joists spanning the 15' distance between interior wooden beam and the outside perimeter beam? Building is 30 feet wide, one story.

Maybe, but that is irrelevant. If the SE designed the foundation and drew up the plans for it, then the inspector is going to want to see the foundation built per the plans. If he sees it done another way, he is going to demand a letter of approval from the SE...why don't you save yourself and especially the SE the hassle associated with that and just ask the SE if it would be faster/cheaper and have him modify the plan?

I'll check with the SE about the modifications.